Problem: Given $ m \angle CBD = 8x - 39$, and $ m \angle ABC = 8x - 69$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since $\angle ABD$ is a straight angle, we know ${m\angle ABD = 180}$ Substitute in the expressions that were given for each measure: $ {8x - 69} + {8x - 39} = {180}$ Combine like terms: $ 16x - 108 = 180$ Add $108$ to both sides: $ 16x = 288$ Divide both sides by $16$ to find $x$ $ x = 18$ Substitute $18$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 8({18}) - 69$ Simplify: $ {m\angle ABC = 144 - 69}$ So ${m\angle ABC = 75}$.